Editing Landau theory (section)

===Applications===
It was known experimentally that the liquid–gas coexistence curve and the ferromagnet magnetization curve both exhibited a scaling relation of the form <math> |T - T_c|^{\beta} </math>, where <math>\beta</math> was mysteriously the same for both systems.  This is the phenomenon of [[Phase transition|universality]].  It was also known that simple liquid–gas models are exactly mappable to simple magnetic models, which implied that the two systems possess the same symmetries.  It then followed from Landau theory why these two apparently disparate systems should have the same critical exponents, despite having different microscopic parameters.  It is now known that the phenomenon of [[Phase transition|universality]] arises for other reasons (see [[Renormalization group]]).  In fact, Landau theory predicts the incorrect critical exponents for the Ising and liquid–gas systems.

The great virtue of Landau theory is that it makes specific predictions for what kind of non-analytic behavior one should see when the underlying free energy is analytic. Then, all the non-analyticity at the critical point, the critical exponents, are because the ''equilibrium value'' of the order parameter changes non-analytically, as a square root, whenever the free energy loses its unique minimum.

The extension of Landau theory to include fluctuations in the order parameter shows that Landau theory is only strictly valid near the critical points of ordinary systems with spatial dimensions higher than 4. This is the [[upper critical dimension]], and it can be much higher than four in more finely tuned phase transitions. In [[Mukhamel]]'s analysis of the isotropic Lifschitz point, the critical dimension is 8. This is because Landau theory is a [[mean field theory]], and does not include long-range correlations.

This theory does not explain non-analyticity at the critical point, but when applied to [[superfluidity|superfluid]] and superconductor phase transition, Landau's theory provided inspiration for another theory, the [[Ginzburg–Landau theory]] of [[superconductivity]].